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Non-integrability criterion for homogeneous Hamiltonian systems via blowing-up technique of singularities
1. | Division of Mathematical Science, Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama-cho, Toyonaka, Osaka 560-8531, Japan |
References:
[1] |
V. I. Arnol'd, Mathematical Methods of Classical Mechanics, $2^{nd}$ edition, Springer, New York, 1989.
doi: 10.1007/978-1-4757-2063-1. |
[2] |
H. Bruns, Über die Integrale des Vielkörper-Problems, Acta Math., 11 (1887), 25-96.
doi: 10.1007/BF02612319. |
[3] |
R. L. Devaney, Triple collision in the planar isosceles three-body problem, Invent. Math., 60 (1980), 249-267.
doi: 10.1007/BF01390017. |
[4] |
R. L. Devaney, Motion near total collapse in the planar isosceles three-body problem, Celestial Mech., 28 (1982), 25-36.
doi: 10.1007/BF01230657. |
[5] |
G. Duval and A. J. Maciejewski, Integrability of Hamiltonian systems with homogeneous potentials of degrees 2. An application of higher order variational equations, Discrete Contin. Dyn. Syst., 34 (2014), 4589-4615.
doi: 10.3934/dcds.2014.34.4589. |
[6] |
S. Kovalevski, Sur le probleme de la rotation d'un corps solide autour d'un point fixe, Acta Math., 12 (1889), 177-232.
doi: 10.1007/BF02592182. |
[7] |
R. McGehee, Triple collision in the collinear three-body problem, Invent. Math., 27 (1974), 191-227.
doi: 10.1007/BF01390175. |
[8] |
R. Moeckel, Heteroclinic phenomena in the isosceles three-body problem, SIAM J. Math. Anal., 15 (1984), 857-876.
doi: 10.1137/0515065. |
[9] |
J. J. Morales-Ruiz, Differential Galois Theory and Non-Integrability of Hamiltonian Systems, Birkhaeuser Basel, 1999.
doi: 10.1007/978-3-0348-8718-2. |
[10] |
J. J. Morales-Ruiz and J. P. Ramis, A note on the non-integrability of some Hamiltonian systems with a homogeneous potential, Methods Appl. Anal., 8 (2001), 113-120. |
[11] |
H. Poincaré, New Methods of Celestial Mechanics Vol. 1, American Institute of Physics, 1993. |
[12] |
M. E. Sansaturio, I. Vigo-Aguiar and J. M. Ferrándiz, Non-integrability of some Hamiltonian systems in polar coordinates, J. Phys. A: Math. Gen., 30 (1997), 5869-5876.
doi: 10.1088/0305-4470/30/16/026. |
[13] |
M. Shibayama, Non-integrability of the collinear three-body problem, Discrete Contin. Dyn. Syst., 30 (2011), 299-312.
doi: 10.3934/dcds.2011.30.299. |
[14] |
M. Shibayama and K. Yagasaki, Heteroclinic connections between triple collisions and relative periodic orbits in the isosceles three-body problem, Nonlinearity, 22 (2009), 2377-2403.
doi: 10.1088/0951-7715/22/10/004. |
[15] |
H. Yoshida, Existence of exponentially unstable periodic solutions and the nonintegrability of homogeneous Hamiltonian systems, Physica, 21 (1986), 163-170.
doi: 10.1016/0167-2789(86)90087-4. |
[16] |
H. Yoshida, A criterion for the nonexistence of an additional integral in Hamiltonian systems with a homogeneous potential, Physica, 29 (1987), 128-142.
doi: 10.1016/0167-2789(87)90050-9. |
[17] |
M. Yoshino, Smooth-integrable and analytic-nonintegrable resonant Hamiltonians, RIMS Kokyuroku Bessatsu, B40 (2013), 177-189. |
[18] |
S. L. Ziglin, Bifurcation of solutions and the nonexistence of first integrals in Hamiltonian mechanics. I. Funktsional. Anal. i Prilozhen., 16 (1982), 30-41, 96. |
[19] |
S. L. Ziglin, Bifurcation of solutions and the nonexistence of first integrals in Hamiltonian mechanics. II. Funktsional. Anal. i Prilozhen., 17 (1983), 8-23. |
show all references
References:
[1] |
V. I. Arnol'd, Mathematical Methods of Classical Mechanics, $2^{nd}$ edition, Springer, New York, 1989.
doi: 10.1007/978-1-4757-2063-1. |
[2] |
H. Bruns, Über die Integrale des Vielkörper-Problems, Acta Math., 11 (1887), 25-96.
doi: 10.1007/BF02612319. |
[3] |
R. L. Devaney, Triple collision in the planar isosceles three-body problem, Invent. Math., 60 (1980), 249-267.
doi: 10.1007/BF01390017. |
[4] |
R. L. Devaney, Motion near total collapse in the planar isosceles three-body problem, Celestial Mech., 28 (1982), 25-36.
doi: 10.1007/BF01230657. |
[5] |
G. Duval and A. J. Maciejewski, Integrability of Hamiltonian systems with homogeneous potentials of degrees 2. An application of higher order variational equations, Discrete Contin. Dyn. Syst., 34 (2014), 4589-4615.
doi: 10.3934/dcds.2014.34.4589. |
[6] |
S. Kovalevski, Sur le probleme de la rotation d'un corps solide autour d'un point fixe, Acta Math., 12 (1889), 177-232.
doi: 10.1007/BF02592182. |
[7] |
R. McGehee, Triple collision in the collinear three-body problem, Invent. Math., 27 (1974), 191-227.
doi: 10.1007/BF01390175. |
[8] |
R. Moeckel, Heteroclinic phenomena in the isosceles three-body problem, SIAM J. Math. Anal., 15 (1984), 857-876.
doi: 10.1137/0515065. |
[9] |
J. J. Morales-Ruiz, Differential Galois Theory and Non-Integrability of Hamiltonian Systems, Birkhaeuser Basel, 1999.
doi: 10.1007/978-3-0348-8718-2. |
[10] |
J. J. Morales-Ruiz and J. P. Ramis, A note on the non-integrability of some Hamiltonian systems with a homogeneous potential, Methods Appl. Anal., 8 (2001), 113-120. |
[11] |
H. Poincaré, New Methods of Celestial Mechanics Vol. 1, American Institute of Physics, 1993. |
[12] |
M. E. Sansaturio, I. Vigo-Aguiar and J. M. Ferrándiz, Non-integrability of some Hamiltonian systems in polar coordinates, J. Phys. A: Math. Gen., 30 (1997), 5869-5876.
doi: 10.1088/0305-4470/30/16/026. |
[13] |
M. Shibayama, Non-integrability of the collinear three-body problem, Discrete Contin. Dyn. Syst., 30 (2011), 299-312.
doi: 10.3934/dcds.2011.30.299. |
[14] |
M. Shibayama and K. Yagasaki, Heteroclinic connections between triple collisions and relative periodic orbits in the isosceles three-body problem, Nonlinearity, 22 (2009), 2377-2403.
doi: 10.1088/0951-7715/22/10/004. |
[15] |
H. Yoshida, Existence of exponentially unstable periodic solutions and the nonintegrability of homogeneous Hamiltonian systems, Physica, 21 (1986), 163-170.
doi: 10.1016/0167-2789(86)90087-4. |
[16] |
H. Yoshida, A criterion for the nonexistence of an additional integral in Hamiltonian systems with a homogeneous potential, Physica, 29 (1987), 128-142.
doi: 10.1016/0167-2789(87)90050-9. |
[17] |
M. Yoshino, Smooth-integrable and analytic-nonintegrable resonant Hamiltonians, RIMS Kokyuroku Bessatsu, B40 (2013), 177-189. |
[18] |
S. L. Ziglin, Bifurcation of solutions and the nonexistence of first integrals in Hamiltonian mechanics. I. Funktsional. Anal. i Prilozhen., 16 (1982), 30-41, 96. |
[19] |
S. L. Ziglin, Bifurcation of solutions and the nonexistence of first integrals in Hamiltonian mechanics. II. Funktsional. Anal. i Prilozhen., 17 (1983), 8-23. |
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